Problem: Given that $3x + y = 10$ and $x + 3y = 14$, find $10x^2 + 12xy + 10y^2$.
Note that \begin{align*}
10x^2 + 12xy + 10y^2 &= (9x^2 + 6xy + y^2) + (x^2 + 6xy + 9y^2) \\
&= (3x + y)^2 + (x + 3y)^2 \\
&= 10^2 + 14^2 = \boxed{296}\end{align*}.